I suggest you take a small piece of the same wood you are using for your CP and drill holes in it and test mount your joystick, see how it feels.
I have routed 1/4" out of 3/4" thick mdf, and on the 1/2" mdf I don't route anything, so I guess my standard is to have JS mounted in a 1/2" thick area, but I cannot say this is the proper or standard way to do it.
It just suites me, and thats what you need to aim for.
I like MDF because not only is it smooth and easy to finish, it is easy to work with and if it says 3/4" thick then it is pretty dad blamed close to 3/4" thick.
Plywood may not be exactly that.
So always measure your material first to make sure you know what you are working with, then to get a feel for what you like, mount the joystick in a piece of scrap and test it for a test drive.